Criteria for Uniqueness and Consistency - Linear Algebra

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}-6&-7&-19\-8&19&-14\0&5&2\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}-21\-36\-149\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}-6&-7&-19\-8&19&-14\0&5&2\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Is there a unique solution for the equation }\&\begin{bmatrix}-20&12\7&16\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-171\149\end{bmatrix}?\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}-20&12\7&16\end{bmatrix}=-404\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}6&-8\3&12\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-188\-101\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}6&-8\3&12\end{bmatrix}=96\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Is there a unique solution for the equation }\&\begin{bmatrix}0&-8&2\-5&3&2\20&12&-14\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}-163\36\22\end{bmatrix}?\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}0&-8&2\-5&3&2\20&12&-14\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Is there a unique solution for the equation }\&\begin{bmatrix}13&13&-2\12&12&12\-16&-16&-11\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}28\125\107\end{bmatrix}?\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}13&13&-2\12&12&12\-16&-16&-11\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}1&11&14\-1&10&-12\11&-5&16\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}59\4\-32\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}1&11&14\-1&10&-12\11&-5&16\end{bmatrix}=-2646\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}9&-12\-3&4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-57\182\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}9&-12\-3&4\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}2&-5&-12\8&7&1\-10&-2&11\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}-71\-38\-50\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}2&-5&-12\8&7&1\-10&-2&11\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}19&-12\9&18\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-126\-163\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}19&-12\9&18\end{bmatrix}=450\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}3&-14&-20\5&0&-3\-6&8&14\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}125\155\-70\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}3&-14&-20\5&0&-3\-6&8&14\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}-16&-4&-9\-13&-13&3\20&-18&9\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}179\-130\-184\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}-16&-4&-9\-13&-13&3\20&-18&9\end{bmatrix}=-4146\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}16&-11\-17&6\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-8\153\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}16&-11\-17&6\end{bmatrix}=-91\&\text{There is a unique solution.}\end{align}$

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